Unified trajectory generation process and system

ABSTRACT

A system, medium, and method including obtaining a plurality of positions for multiple components defined by a plan; obtaining a set of constraints that express limitations for the multiple components at the plurality of positions, the constraints being applicable to a plan where the multiple components synchronously change their positions with time to traverse a prescribed sequence of the plurality of positions, at least one of the multiple components being further constrained to change its position over time by staying within a predefined tolerance to a predefined smooth function of position over time between different positions; determining a trajectory of position and a minimum duration in which the multiple components completely synchronously traverse the prescribed sequence of positions while satisfying the constraints for the multiple components; and generating a record of the determined trajectory of position and the minimum duration for the plurality of components.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 15/182,080, filed on Jun. 14, 2016, the contents ofwhich are hereby incorporated by reference.

BACKGROUND FIELD

The embodiments described below generally relate to controlling aradiation therapy delivery system. More specifically, some embodimentsrelate to generating a unified trajectory for a radiation therapydelivery system and controlling the system in accordance therewith.

DESCRIPTION

Contemporary radiotherapy treatment planning systems may providespecialized versions of step-and-shoot intensity modulated radiotherapy(IMRT) and arc based IMRT. Some notable examples are Varian's RapidArc(Varian Medical Systems, Palo Alto, Calif., USA), Elekta's VMAT (ElektaLtd, Crawley, UK), and Siemens' Modulated Arc (mARC) (Siemens AGHealthcare, Erlangen, Germany). In some aspects, during the treatmentplanning process parameters such as gantry speed, leaf speed, dose rate,and the limits on the bounds of such quantities are considered. Thetreatment plans are then passed to a linac (i.e., a linear acceleratorof the radiation treatment delivery system) controller as a collectionof discrete control points. Each of the control points describes thesimultaneous position of each mechanical component of the linac, as wellas the cumulative delivered dose of radiation up to that instant.

When the radiation treatment delivery system is in an intermediate statebetween control points, it is implicitly assumed that the components ofthe delivery system move with a constant velocity that results in alinear trajectory of movement between the control points. As usedherein, a linear trajectory implies a one dimensional or angularposition trajectory that is linear over time. It is not astraight-forward process to determine the delivery parameters, as afunction of time, that would ensure that the implicit piecewise lineartrajectory of the radiation delivery system components in between thecontrol points is faithfully executed, while not prohibitivelyprolonging the delivery process. This may be due, at least in part,because an ideal piecewise linear trajectory between successive“segments” will require infinite acceleration at the control point.Otherwise, one can use a very small acceleration to achieve a lowconstant velocity that will ensure linearity but at the cost ofincreasing delivery time.

Additionally, this is not a trivial concern since, for example,specification of a linear trajectory between two points for onecomponent may result in longer delivery times for the system as a whole.

Conventionally, the delivery parameters may be decided locally forindividual segments, as opposed to optimum delivery parametersconsidering the whole trajectory. For example, for a particular type oftreatment the controller may attempt to deliver the treatment of eachsegment (or partial arc) with a constant dose rate and constant gantryspeed. The linked delivery parameters (e.g., cumulative output monitorunits (MU), radiation output rate and the gantry speed, etc.) of eachsegment are pre-computed using, for example, the following heuristicsof:

1. If the MU value is large, then the controller tries to deliver thetreatment plan using the maximum dose rate while varying gantry speed.

2. When a smaller number of MUs are to be delivered, the controllertries to use the maximum gantry speed while varying the dose rate.

3. For intermediate MU ranges, the controller varies both the dose rateand the gantry speed. In all cases, the chosen gantry speed and the doserate are chosen in such a way that the resultant duration is sufficientfor the multileaf collimator (MLC) or other beam-shaping device(s) tochange their shape to accommodate all segments.

In one previous attempt to obtain the optimum delivery parameters thatmight result in a minimum beam-on time by establishing theirinterrelationships and solving an optimization problem, neither theacceleration and the jerk capability of the system nor the constraintson the different components are explicitly treated or modeled by thedelivery system vendors. As a result, the smoothness of the treatmentdelivery cannot be guaranteed. Instead, the gantry position, the leafpositions, and the cumulative dose output by the system are monitoredperiodically (e.g., every 50 ms in Varian's RapidArc) and onlineadjustments are made in order to try to stay close to the piecewiselinear trajectory of the treatment plan.

In much of the previous work regarding the trajectory of roboticmanipulators, splines and variants thereof are used as trajectorymodels. While spline and spline-like functions might succeed atmaintaining via-point accuracy and providing smooth trajectories, thesefunctions tend to ripple (oscillate) and depart significantly from alinear trajectory. In some regards, a constrained cubic spline methodhas been developed that attempts to tackle an overshoot problem.However, constrained splines result in discontinuous acceleration(s) atvia points. Additionally, attempts to realize such non-physicaltrajectories may lead to excessive mechanical wear and excitation ofhigher system modes that lead to further inaccuracies.

A trapezoidal velocity model has been widely used in industry forstandard position control of some mechanical components. However, such amodel requires discontinuous acceleration and infinite jerk to realizetrajectories. In some instances, instead of a spline based model forposition or a trapezoidal velocity model, an exponential velocity model(EVM) has been used to obtain a trajectory for a single component thatpasses through multiple via points, while maintaining velocity,acceleration, and jerk constraints. The EVM has some advantage over thetrapezoidal velocity model owing to the continuous nature of itsvelocity, acceleration and jerk profiles. Also, unlike commonly employed3rd-order spline based position models, it exhibits less over- andundershoot, and realizes a continuous jerk profile. However, somedisadvantages of the EVM include, for example, the computed trajectoriesare unnecessarily slow since maximum acceleration and jerk are notmaintained for as long as possible, and the position trajectories cannotbe expressed in an analytical closed form. An infinite series expressionis, however, available for the position trajectory. In some aspects,time-optimal velocity profiles are constructed for each “segment” andthe velocity profiles are then modified so that the final velocity of asegment blends into the initial velocity of the next segment. However,no attempt is made to optimize the total time or to keep the positiontrajectory within tolerance of a predefined specification thereof.

Improved control of a radiation treatment delivery system to provide anoptimal delivery duration and close observance to the linear trajectoryof a treatment plane is desired, with all of the constraints of positionfor the radiation treatment delivery system being considered andsatisfied.

The above arguments apply also to diagnostic imaging systems. In suchsystems, one or more sources of ionizing and non-ionizing radiation maybe required to move synchronously with one or more imaging detectors,and also the patient support (couch, table, platform or seat). Usually,the synchrony must also include the delivered imaging radiation rate andamount. The teachings contained herein apply equally to such systems. Ingeneral, references to the linear accelerator, the radiation source,apply also to radioisotope sources, diagnostic imaging x-ray tubesources, sources of ultrasound, and electromagnetic fields such as thoseused in magnetic resonance imaging.

SUMMARY

In order to address the foregoing, some embodiments provide a system,medium, and method including obtaining a set of position relatedconstraints for a plurality of mechanical and radiation producingcomponents of a radiation treatment delivery system, the constraintsapplicable to the radiation treatment delivery system delivering theradiotherapy treatment plan by synchronous motion of the plurality ofmechanical and radiation producing components traversing a prescribedsequence of the plurality of positions in a predetermined time;determining a trajectory and a minimum duration for the radiationtreatment delivery system to traverse the prescribed sequence of theplurality of positions and deliver the radiotherapy treatment plan whileadhering to the constraints and being within a predetermined tolerancelimit of a linear trajectory along the prescribed sequence of theplurality of positions; and controlling the plurality of mechanical andradiation producing components of the radiation treatment deliverysystem to execute the determined trajectory within the determinedminimum duration, as indicated in the record.

The appended claims are not limited to the disclosed embodiments,however, as those in the art can readily adapt the descriptions hereinto create other embodiments and applications.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments will become readily apparent from consideration of thefollowing specification as illustrated in the accompanying drawings, inwhich like reference numerals designate like parts, and wherein:

FIG. 1 is a perspective view of a radiation treatment room, compatiblewith some embodiments;

FIG. 2 is a depiction of a trajectory, illustrating some aspects of aradiotherapy treatment plan;

FIG. 3 is an illustrative depiction of a generic class of velocitymodel, in accordance with some embodiments herein;

FIG. 4 is an illustrative depiction of a representation of kinematicconstraints of components of a radiation delivery system, in accordancewith some embodiments;

FIG. 5 is an illustrative depiction of some aspects of a step-and-shoottreatment plan, in accordance with some embodiments

FIG. 6 is an illustrative depiction of some aspects of a step-and-shoottreatment plan, in accordance with some embodiments; and

FIG. 7 is an illustrative depiction of a position and velocity profiles,according to some embodiments herein

DETAILED DESCRIPTION

The following description is provided to enable a person in the art tomake and use some embodiments and sets forth the best mode contemplatedby the inventors for carrying out some embodiments. Variousmodifications, however, will remain readily apparent to those in theart.

FIG. 1 illustrates radiotherapy treatment room 100 that may provide acontext for some aspects, contexts, use-cases or platforms of thepresent disclosure, according to some embodiments. Radiotherapytreatment room 100 includes linear accelerator (linac) 105, table 155and operator console 160. The various components of radiotherapytreatment room 100 may be used to deliver a beam of radiation to atarget volume such as phantom 180. The target volume may comprise apatient positioned to receive the beam according to a radiationtreatment plan. The elements of treatment room 100 may be employed inother applications according to some embodiments.

Linac 105 generates and emits a radiation beam (e.g., an x-ray beam)from treatment head 110. More particularly, the radiation originates atradiation source 150. The radiation beam may comprise electron, photonor any other type of radiation. According to some embodiments, the beamexhibits energies in the megavoltage range (i.e. >1 MeV) and maytherefore be referred to as megavoltage beam.

Treatment head 110 is coupled to a projection of gantry 115. Gantry 115is controllable to be rotatable around gantry axis 120. As indicated byarrow 125, gantry 115 may rotate clockwise or counter-clockwiseaccording to some embodiments. Rotation of gantry 115 serves to rotatetreatment head 110 around axis 120.

During radiation emissions (e.g., treatment, calibration, and otherprocedures) treatment head 110 emits a divergent beam of megavoltagex-rays along beam axis 130. The beam is emitted towards isocenter 135 oflinac 105. Isocenter 135 may be located at the intersection of beam axis130 and gantry axis 120. Due to divergence of the beam and the shapingof the beam by beam-shaping devices in treatment head 110, the beam maydeliver radiation to a volume of phantom 180 rather than only throughisocenter 135.

Table 155 may support a patient during radiation treatment and supportphantom 180 during aspects discussed herein. Table 155 may be adjustableto assist in positioning phantom 180 or a particular target volume of apatient at isocenter 135. Table 155 may also be used to support devicesused for such positioning, for calibration and/or for verification.

Imaging device 140 may comprise any system to acquire an image based onreceived radiation. Imaging device 140 may be attached to gantry 115 inany manner, including an extendible and retractable housing 118.Rotation of gantry 115 may cause treatment head 110 and imaging device140 to rotate around isocenter 135 such that isocenter 135 remainslocated between treatment head 110 and imaging device 140 throughoutstationary and rotational movements of gantry 115.

Imaging device 140 may acquire projection images before, during and/orafter radiation treatment. In some embodiments, imaging device 140 mayinclude an analog or a digital radiation detector. Imaging device 140may be used to acquire images based on radiation emitted from treatmenthead 110. These images may reflect the attenuative properties of objectslocated between treatment head 110 and imaging device 140.

Radiation source 150 may include any sources to emit kilovoltageradiation or other imaging radiation that are or become known. In someembodiments, radiation source 150 may employ a cathode based on carbonnanotube or thermionic emission technology. In some embodiments, eachradiation source 150 may be disposed in a fixed relationship withrespect to treatment head 110.

It is noted that the source-detector trajectory of the radiotherapysystem including the imaging system may be controlled, in someembodiments, to move in a circular or non-circular trajectory and/orhave a fixed or variable radiation source—detector distance.

Operator console 160 includes input device 165 for receivinginstructions from an operator such as an instruction to calibrate linearaccelerator 105 and an instruction to move radiation source 150 in aparticular trajectory and to deliver radiation from a number of discretepositions or locations along the trajectory path, according to atrajectory path of a radiotherapy treatment plan and a velocity model.Console 160 also includes output device 170 that may include a monitorfor presenting calculated projection images, acquired projection images,three-dimensional images, operational parameters of linear accelerator105 and/or interfaces for controlling elements thereof. Input device 165and output device 170 are coupled to processor 175 and storage 180.

Processor 175 executes program instructions or code according to someembodiments. The program instructions may be executable to controllinear accelerator 105 to operate as described herein. The programinstructions may be stored in storage 180, which may comprise one ormore tangible storage media of identical or different types, includingbut not limited to a fixed disk, a floppy disk, a CD-ROM, a DVD-ROM, anoptical disk or storage device, a magnetic tape, a solid state storagedevice, a flash drive, and a other devices and systems. Storage 180 maystore, for example, virtual models of phantoms, initial imaging geometryparameters, radiation treatment plans, set(s) of position relatedconstraints for the plurality of mechanical and radiation producingcomponents of the radiation treatment system 105, software applicationsto control (e.g., calibrate, etc.) linear accelerator 105 and/or toprovide radiation treatment, and other data used to perform radiationtreatment.

Operator console 160 may be located apart from linear accelerator 105,such as in a different room, in order to protect its operator or otherentity from radiation. For example, linear accelerator 105 may belocated in a heavily shielded room, such as a concrete vault, whichshields the operator from radiation generated by accelerator 105.

Each of the devices shown in FIG. 1 may include fewer, more, ordifferent elements than those shown and are not limited to the devicesshown in FIG. 1.

In external beam radiotherapy, treatment delivery is modeled as thesynchronous actuation of multiple mechanical and radiation producingcomponents through a prescribed sequence of discrete states, also knownas “control points”. In-between the control points, the motion isconstrained but unspecified and depends upon the implementation ofspecific delivery modes (e.g., step-and-shoot, modulated arc, etc.). Thepresent disclosure includes processes to generate a continuous positionversus time trajectory that enables direct control of the trade-offbetween delivery accuracy and delivery time.

The present disclosure introduces a number of innovations to achievesuch an optimized trajectory. In some aspects, the processes hereininclude, for any component that follows a general class of onedimensional (1-D) velocity models containing three phases (monotonicincrease (or decrease), constant velocity, and monotonic decrease (orincrease)) of velocity, a process to determine the minimum duration, aswell as the feasibility of a given duration for a position trajectory ofsatisfying three sets of constraints for the mechanical and radiationproducing components of the radiation treatment system is presentedherein.

A first set of constraints, Set I, includes boundary conditions, as wellas constraints on total execution time and distance, including an entryvelocity, an entry acceleration (zero), an exit velocity, an exitacceleration (zero), a time taken to execute the trajectory, and adistance between entry and exit positions.

A second set of constraints, Set II, may include derivatives of theposition of the mechanical and radiation producing components of theradiation treatment system during the trajectory. Set II can include theconstraints of a jerk parameter, an acceleration parameter, and avelocity.

A third set of constraints, Set III, can include and consider trajectorytolerance(s) that constrain a generated position trajectory to staywithin a pre-defined tolerance limit of the linear position trajectoryof the radiation treatment plan over time between two positions (i.e.,control points). If such a trajectory is feasible, then a process hereinmay be used to determine a cruise velocity and the durations for theacceleration, cruise, and deceleration intervals of the trajectorybetween the two positions.

In some aspects herein, in response to a need to enforce the constraintconditions of Sets I-III, an optimization process is disclosed todetermine the position trajectories of, for example, linear accelerator(linac) components (including machine radiation output rate and itsderivatives) with a goal to minimize a total duration of delivery whilesatisfying all of the constraints.

In some embodiments, in the instance of an interruption of a treatmentdelivery, a process for the fast computation of a resumption trajectorythat catches up with the calculated/determined optimized trajectorywithin few control points is disclosed herein. Accordingly, theinterrupted calculated/determined optimized trajectory may be resumedwithin traversing a few control points.

In some aspects herein, the present disclosure demonstrates a possibletrade-off between delivery accuracy and delivery time that may be usedto speed-up delivery of conventional step-and-shoot treatments byallowing some mechanical components to move during short intervalswithin a prescribed tolerance (as opposed to being static as is the casein conventional step-and-shoot contexts).

Accordingly, optimized trajectories have been realized by using theprocesses disclosed herein, with the results being confirmed bycomparison to, for example, the average delivery times reported by theradiation treatment delivery system's manufacturer. In some instances, aresumption trajectory was computed from the point of interruption thatenabled a resumption of the original trajectory (i.e., catch-up) within,for example, 4 control points. In other aspects, Applicant has realizedin the context of a step-and-shoot treatment plan with, for example, 71segments, how one can generate a trajectory that reduces a planneddelivery time by about 15% by allowing the gantry to move within sometolerance during the shoot segments.

In some aspects, the present disclosure presents a framework andprocesses to achieve fast synchronized movement of multiple mechanicalcomponents via predefined positions (i.e., control points) whilesatisfying all mechanical constraints and without significant reductionin positional accuracy. The presented framework, when applied inradiotherapy treatment delivery, also allows the designers and otherentities to implement original and hybrid therapy modes without the needto develop specific automation strategies.

In some aspects, the problem of determining the delivery parameters maybe approached from the perspective of trajectory generation for roboticmanipulators. In robotics, the generation of the time-optimumsimultaneous trajectory of multiple manipulators between “via points”,while respecting velocity, acceleration, and jerk constraints, is wellstudied for certain specific models of position and velocity. In theproblem formulation herein, we identify a via point of a roboticmanipulator end-point trajectory with a control point of a radiotherapydelivery. Some of the methods presented herein obtain the trajectory ofdelivery system components by formulating a trade-off between theoverall delivery time and the degree to which the generated trajectoryconforms to a model piecewise linear trajectory. Presented hereinbelowis a brief review of some aspects of trajectory generation for roboticmanipulators and a discussion of its limitations to the problem oftrajectory generation of radiation therapy delivery system (RTDS)components during treatment delivery that are more broadly addressedherein.

Some of the shortcomings of some other approaches regarding trajectorygeneration for a RTDS (radiation therapy delivery system) may includeemphasizing generating time-optimal trajectories and via-point accuracywithout constraining trajectories to remain within a tolerance of somepredefined (e.g., linear) trajectory in between via points. FIG. 2 is anillustrative depiction of a trajectory 200 including a plurality of viapoints such as, for example, points 205 and 210. The trajectory model isshown as comprising a piece-wise linear trajectory at 215. The actualtrajectory is shown at 220. Seen in FIG. 2 is the deviation ordifference between the piecewise linear trajectory model's profilebetween the via points and the actual robotic motion shown at 220.

In some aspects, each of the spline, trapezoidal, and EVM approaches aretied to some specific position or velocity model. As such, these andother approaches do not provide a generic framework that is applicableto a large class of velocity (and acceleration and jerk) models formoving linked components that obtains a simultaneous time-optimaltrajectory for the components, while remaining within a certaintolerance of some specified trajectory.

In some embodiments, the present disclosure addresses some of theinadequacies of other approaches and provides a trajectory generationframework involving a number of unique features. These features include,for motion between two 1-D positions involving a generic class ofvelocity model that contains an interval of monotonic increase (ordecrease), a constant velocity interval, and an interval of monotonicdecrease (or increase, respectively). A process is disclosed that candetermine both a minimum duration, as well as the feasibility of a givenduration for a position trajectory satisfying a plurality (i.e., set) ofconstraints. In some embodiments, the set of constraints include anentry and exit velocity; the predefined duration and distance betweenthe two points; an entry and exit acceleration that are both set tozero; and jerk, acceleration, and velocity parameters that areconstrained (i.e., bound). Furthermore, the generated positiontrajectory stays within a predefined tolerance limit of the lineartrajectory over time. If such a trajectory is feasible, then a processherein provides a mechanism to determine a cruise velocity and durationsfor the acceleration, cruise and deceleration intervals.

Regarding the generic class of velocity model, a number of differentvelocity models, including but not limited to, a trapezoidal velocitymodel, a linear position trajectory with polynomial blend, a double-Svelocity model, and a 15-segment trajectory or EVM are within thiscategory.

The present disclosure also relates to and presents, using theprocess(es) introduced above as a basis, a process to determine 1-Dposition trajectories of the mechanical components and the dose of anygeneric radiotherapy treatment plan such that the components movesynchronously between consecutive control points while respecting allconstraints on position, velocity, acceleration and jerk whileminimizing the total duration of delivery.

Some aspects herein relate to a scenario wherein the delivery of atrajectory optimized as disclosed herein is interrupted. In such cases,the present disclosure provides a process for a fast computation of aresumption trajectory that returns or catches up with the pre-computedoptimized trajectory within few control points. Therefore, although theoptimized trajectory execution may be interrupted, the system herein mayquickly return to the optimized trajectory based on the calculation andexecution of the resumption trajectory.

Some embodiments herein further relate to using at least some of theprocesses disclosed herein to reduce the delivery time associated withstep-and-shoot delivery treatments, without undue sacrifice of deliveryaccuracy. A characteristic of step-and-shoot treatment deliveries isthat the mechanical components remain static during an interval when thebeam is on (i.e., during shoot segment). The present disclosure providesa mechanism to generate a trajectory with a reduced delivery time byallowing a gantry, for example, to move within some predefined toleranceduring the shoot segments.

In some embodiments, each element of an RTDS that is implicated in acontrol point, including the radiation output itself, may be consideredas a component. The individual discrete positions of each component areprescribed by the treatment plan at the control points. Between eachcontrol point, each component moves in a linear or circular pathfollowing a “change-cruise-change” velocity profile or model. Yet, acontinuous position versus time trajectory of any of the components isnot prescribed in the treatment plan. However, for any component and forany segment, the actual trajectory is expected to be close to linear(over time), and the deviation from a linear trajectory is constrained(i.e., limited). There also exist constraints on velocity, acceleration,and jerk specific to each component and at each segment. Accordingly, agoal of some embodiments herein is to generate synchronous trajectoriesof all the components in such a way that the total delivery time isminimized and all of the constraints are satisfied.

For purposes of some embodiments herein, it is assumed that a radiationtreatment plan is described based on the International ElectrotechnicalCommission (IEC) and Digital Imaging and Communications in Medicine(DICOM) standards, unless otherwise stated. Also, the notations definedin Table 1 below may be used throughout the following discussion.

In some aspects, the following pre-processing steps may be applied tothe treatment plan before a trajectory generation process is performed.For any rotating component (e.g., gantry, collimator, etc.), as long asthe rotation direction is not reversing, the angle values for thecomponents in the control points are re-expressed in a contiguousmonotonic fashion so that an algebraic subtraction of the angle valuesin consecutive control points reveal the amount of rotation. Also, thetreatment plan may only specify the positions of the leaf end points.Multiple combinations of carriage positions and leaf extents thatsatisfy Equation 1 (described in Table 1 below) may achieve the sameposition of the leaf end points. Unless a treatment plan explicitlydescribes the carriage positions, heuristics are used to decompose thepositions of the leaf end points into a “suitable” combination of theposition of the common carriage and individual leaf extents. Thisapproach facilitates the treatment of the carriages and the leaf extentsas separate “components” in the trajectory generation process(es)herein.

A RTDS may have many different moving components that each haveindividual limits on their position, velocity, acceleration, jerk andother motion parameters. Successful delivery of a radiotherapy planrequires simultaneous movement of these components through a series ofconfigurations. For accurate delivery, the components are (implicitly)expected to travel at somewhat constant velocity between respectivesuccessive configurations. To ensure rapid plan delivery, the componentsare expected to move between configurations as fast as possible withoutviolating any limits/constraints, all the while maintaining synchronousarrival at each control point. These twin objectives may be describedbelow in a formal way, as illustrated in Table 1.

TABLE 1 Notations for RTDS components θ_(G), Θ_(MLC), Φ_(C), X_(C),Y_(C), Z_(C): the angular position of the RTDS gantry, the multi-leafcollimator (MLC) and the couch, and the lateral, longitudinal andvertical positions of the couch respectively. Y_(J1), Y_(J2): Positionof the Y1 and Y2 jaw beam limiting device (BLD). X_(l11), . . . ,X_(l1N), . . . , and X_(l21) , . . . , X_(l2N),: Positions of the endpoints of X1 and X2 leaves of the MLCs as specified in the plan. The MLCleaves are separated in two banks, N leaves in each bank and each bankis connected to a carriage. The positions of the leaf end points aredetermined by the position of the common carriage and the extent bywhich the individual leaves extend from the carriage. X_(B1),X_(B2):Position of the X1 and X2 bank (carriage). X_(L11), . . . , X_(L1N), . .. , and X_(L21), . . . , X_(L2N): The non-negative extents by whichindividual leaves protrude from their respective carriages. The leavesconnected with the X1 (respectively, X2) carriage can only extend inpositive (respectively, negative) X direction in BLD system. X_(B1)^(min), X_(B1) ^(max), X_(B2) ^(min), X_(B2) ^(max): These are theallowable minimum and maximum postion of the X1 and X2 carriages in theBLD co-ordinate system. X_(L1) ^(min), X_(L1) ^(max), X_(L2) ^(min),X_(L2) ^(max): These are the minimum and the maximum (non-negative)extent of the leaves connected to X1 and X2 carriages. The followingrelationships hold ∀1 ≤ k ≤ N: X_(l1k) = X_(L1k) + X_(B1); X_(L1) ^(min)= 0 ≤ X_(L1k) ≤ X_(L1) ^(max); X_(B1) ^(min) ≤ X_(B1) ≤ X_(B1) ^(max)X_(l2k) = X_(B2) - X_(L2k); X_(L2) ^(min) = 0 ≤ X_(L2k) ≤ X_(L2) ^(max);X_(B2) ^(min) ≤ X_(B2) ≤ X_(B2) ^(max) (1)

 : The cumulative dose Ξ = {

 , θ_(G), Θ_(MLC), Y_(J1), Y_(J2), X_(l11), X_(l12), • • • X_(l1N),X_(l21), X_(l22), • • • X_(l2N), Φ_(C), X_(C), Y_(C), Z_(C)} is the setof all components. The treatment plan is a sequence of beams. Each beamis a sequence of control points. CP^(j) = [

 ^(j), θ_(G) ^(j), Θ_(MLC) ^(j), Y_(J1) ^(j), Y_(J2) ^(j), Y_(l11) ^(j),X_(l12) ^(j), • • • X_(l1N) ^(j), X_(l21N) ^(j), X_(l22N) ^(j), • • •X_(l2N) ^(j), Φ_(C) ^(j), X_(C) ^(j), Y_(C) ^(j), Z_(C) ^(j)] is aspecification of each compoinent ξ ϵ Ξ in the j-th control point. Atreatment plan is a sequence of M + 1 control points CP^(j), 0 ≤ j ≤ M.In particular,

 ^(j) is the cumulative dose value of control point CP^(j) such that

 ^(j) = 0 and

 ^(j+1) ≥

 ^(j). All type of treatments (step-and shoot, dynamic) are captured bythe above description. The transistions between the control points arecalled segments and are numbered by the index s. In a treatment plan ofM + 1 control points, there will be M segments where segment s, 1 ≤ s ≤M is a transition from CP^(s−1) to CP^(s). T⁰ denotes the start time ofdelivery, and T^(j) denotes the time when the control point CP^(j) isreached simultaneously by all components. ξ(t), ξ(CP^(j)): For eachcomponent ξ, let ξ(t) denote the position (or cumulative dose in case ofthe ″dose″ component) of a component at time t in the generatedtrajectory. Also let ξ(CP^(j)) denote the position of the component ξ atcontrol point j as prescribed by the treatment plan. V_(max) ^(ξ,s) ,A_(max) ^(ξ,s) , J_(max) ^(ξ,s): the absolute limit on velocity,acceleration and jerk, respectively, for any component ξ at the segments. Due to the effect of gravity, the limits may be different dependingon the projection of the gravitational field along the direction ofmotion. To accomodate this, the constraint limits are segment-dependent.${\xi_{linear}(t)}:={{\xi\left( {CP}^{j} \right)} + {\frac{\left( {{\xi\left( {CP}^{j + 1} \right)} - {\xi\left( {CP}^{j} \right)}} \right)}{\left( {T^{j + 1} - T^{j}} \right)} \cdot \left( {t - T^{j}} \right)}}$denotes the ideal trajectory of each component that is linear over timebetween successive control point positions. And Δ^(ξ) denote the maximumdeviation from this ideal linear trajectory (as shown in Figure 2) thatmay be tolerated for the component ξ. This limit is determined duringtreatment planning.

Given the set of mechanical (and dose) components Ξ, a treatment planconsisting of M+1 control points (M segments), a velocity model, limitson the velocity, the acceleration and the jerk for each component ξ∈Ξ ineach segment s|1≤s≤M, we want to find a sequence of time points 0=T⁰≤T¹≤. . . T^(M) such that the total delivery time is minimized whilesimultaneously satisfying the following constraints (A1, A2, and A3).

For constraint A1, each component moves simultaneously between theirrespective control point positions while their individual trajectoryover time is close to linear trajectory over time. That is,∀ξ,∀j,0≤j≤M, ∀t,T ^(j) ≤t≤T ^(j+1), ξ(t=T ^(j))=ξ(CP^(j)) and|ξ(t)−ξ_(linear)(t)|≤Δ^(ξ)  (2)

For constraint A2, all of the components begin (at first control point)and end (at last control point) their trajectories with zero velocitiesand accelerations. That is,∀ξ,{dot over (ξ)}(t=T ⁰)={dot over (ξ)}(t=T ^(M)=0 and {umlaut over(ξ)}(t=T ⁰)={umlaut over (ξ)}(t=T ^(M))=0  (3)For constraint A3, for each component

ξ and for each segment s, all the constraints on velocity, accelerationand jerk (J_(max) ^(ξ,s)), are satisfied:∀ξ,∀s1≤s≤M,j=s−1, and ∀t,T ^(j) ≤t≤T ^(j+1), |{dot over (ξ)}(t)|≤V_(max) ^(ξ,s)|{umlaut over (ξ)}(t)|≤A _(max) ^(ξ,s), and |ξ(t)|≤J _(max)^(ξ,s)  (4)where j is the control point index and s is segment index.

In some embodiments, a desired outcome of a trajectory generation hereinrequires that the total delivery time is minimized while satisfying allof the foregoing constraints. For a segment s between the control pointsj and j+1, let h^(s)=T^(s)−T^(s−1)=T^(j+1)−T^(j) be the time taken tomove all components as planned. Then, the objective is to minimize thetotal delivery time that can be represented as:T _(del):=Σ_(s=1) ^(M) h ^(s)  (5)

In order to achieve the above aspects, a number of conditions need toexist. One is a process to determine the feasibility of a trajectorythat describes the motion of any component between two positions (i.e.,control points) in one dimension, in a given duration and with givenentry and exit speeds while following a general class of “monotonicchange—cruise—monotonic change” velocity models, and satisfyingposition, velocity, acceleration and jerk limits. Another condition isan optimization process that iteratively allocates durations to eachsegment, uses the above process to check the feasibility of theallocated durations for each component and each segment, and determinesthe feasible allocation that results in the minimum total duration.

Regarding the feasibility of a position trajectory between two pointsfor a generic class of velocity models, a process is undertaken todetermine whether a component can traverse a given distance in a giveninterval while respecting kinematic constraints and following a generalclass of velocity model involving “monotonic change—cruise—monotonicchange” pattern. Such a generic class of velocity model is illustratedin FIG. 3. As shown in FIG. 3, the velocity model 300 exhibits a“monotonic change—cruise—monotonic change” behavior in the form of amonotonic increase in velocity from an entry velocity 305 to a cruisevelocity 310, a cruise phase 312, and a monotonic decrease from thecruise velocity 310 to the exit velocity 315. A settling time 320 fromthe entry velocity to the cruise velocity is shown, as well as asettling time 325 from the cruise velocity to the exit velocity. Thetotal duration, h, is the sum of the two settling times and the time tocruise.

To determine the feasibility of a trajectory for the generic class ofvelocity model that satisfies the constrains and limits as discussedabove, two sub-problems are defined and addressed. In the first, theminimum duration that is required to travel a certain distance under thegiven kinematic constraints is determined. In the second, adetermination is made of whether it is feasible to cover a certaindistance within a given duration under the given kinematic constraints,and to calculate the durations of the acceleration, cruise, anddeclaration phases. A formal description of these problems follows.

It is noted that the feasibility determinations presented herein may berelevant to many applications, of which radiotherapy is just one. Whilesome of the processes disclosed herein are meant for objects moving in astraight line, they may be equally applicable for objects moving incircular trajectories, where the linear position and its derivatives maybe respectively replaced by quantities relevant to rotational motion.

In some aspects we can assume an object travels in a straight line frompoint P_(entry) to point P_(exit). Without any loss of generality let usassume that, along the motion axis, P_(exit)>P_(entry). Given thedistance D=|P_(exit)−P_(entry)|, monotonic continuous velocity modelsf(⋅) and g(⋅), the allowable maximum absolute speed V_(max), the entryspeed V_(entry)≤V_(max), the exit speed V_(exit)≤V_(max), the maximumabsolute acceleration A_(max), the maximum absolute jerk J_(max) and aposition tolerance Δ, we want to find the minimum duration h and itscomponents, namely the entry settling time T_(en), the exit settlingtime T_(ex), the cruise duration T_(cruise), and the cruise velocityV_(cruise)≤V_(max), such that the following conditions (1-5) aresimultaneously satisfied. Here the entry settling time (respectively theexist settling time) is the time required to change the speed of thecomponent from the entry speed to a cruise speed (respectively from thecruise speed to the exist speed), and cruise speed is the constant speedwhere the component travels without any acceleration or deceleration.

Per Condition 1, the velocity model of the object follows a monotonicchange—cruise—change pattern as shown in FIG. 3 and as expressed below:

                                       (6) ${v(t)}:=\begin{Bmatrix}{{{f(0)} = {{V_{{entry},}{\overset{.}{f}(0)}} = 0}},} & {{{if}\mspace{14mu} t} = T_{entry}} \\{f\left( {t - T_{entry}} \right)} & {{{if}\mspace{14mu} T_{entry}} < t < {T_{entry} + T_{en}}} \\{{f\left( T_{en} \right)} = {V_{cruise} = {g\left( T_{ex} \right)}}} & \; \\{{{and}\mspace{11mu}{\overset{.}{f}\left( T_{en} \right)}} = {{\overset{.}{g}\left( T_{ex} \right)} = 0}} & {{{{if}\mspace{14mu} T_{entry}} + T_{en}} \leq t \leq {T_{exit} - T_{ex}}} \\{g\left( {T_{exit} - t} \right)} & {{{{if}\mspace{11mu} T_{exit}} - T_{ex}} < t < T_{exit}} \\{{{g(0)} = V_{exit}},{{\overset{.}{g}(0)} = 0}} & {{{if}\mspace{14mu} t} = T_{exit}}\end{Bmatrix}$

For Condition 2, the total duration is composed of the entry settlingtime, the cruise time and the exit settling time, as represented below.h=T _(exit) −T _(entry) =T _(en) +T _(cruise) +T _(ex)  (7)

The Condition 3, states that the limits on the velocity, accelerationand jerk are to be satisfied:|v(t)|≤V _(max) , |a(t)={dot over (v)}(t)|≤A _(max), and |J(t)={umlautover (v)}(t)|≤J _(max)  (8)

Condition 4 states that the prescribed distance D is to be traversed inthe duration, as represented below:∫_(T) _(entry) ^(T) ^(entry) ^(+T) ^(en) f(t−T _(entry))dt+V _(cruise)·T _(cruise)+∫_(T) _(exit) _(−T) _(ex) ^(T) ^(exit) g(T _(exit)−t)dt=D  (9)

For Condition 5, the deviation of the generated position trajectory fromthe linear position trajectory over time has to stay within toleranceand is represented as follows:

$\begin{matrix}{{{{{{q(t)} - {q_{linear}(t)}}} \leq {\Delta{\forall t}}},{T_{entry} \leq t \leq T_{exit}}}\mspace{14mu}{{{{where}\mspace{14mu}{q(t)}}:={\int_{T_{entry}}^{t}{{v(t)}{dt}}}},{{q_{linear}(t)}:={\frac{D}{h} \cdot \left( {t - T_{entry}} \right)}}}} & (10)\end{matrix}$

With regards to determining the minimum duration h and its components,three sub-procedures may be used to solve this problem. (1) Adetermination is made to find out how much settling time is needed tochange the velocity of an object from one velocity to another velocityunder the given constraints. (2) A determination is made regarding thedistance that is traversed during this change in velocity. And, (3) adetermination is made regarding the time instant where the generatedposition trajectory is maximally different from a linear positiontrajectory.

Regarding sub-procedure 1, given the monotonic velocity model f(⋅) withknown initial velocity and zero initial acceleration (f(0)=V_(i), {dotover (f)}(0)=0), we find the settling time T_(settling) to reach somecruise velocity with zero acceleration (f(T_(settling))=V_(f), {dot over(f)}(T_(settling))=0). This settling time depends on the specificfunctional form and parameters of the velocity model. Specific solutionsfor exponential and double-S velocity models are presented in theAppendix. In general, one finds the time T_(mid) by which the velocitychanges from V_(i) to the mid-velocity |(V_(f)+V_(i))/2| and theacceleration changes from 0 to a maximum value. The maximum value of theacceleration is limited by A_(max), the maximum value of the jerk islimited by J_(max), and whenever the jerk becomes zero when theacceleration reaches the maxima at this mid velocity position. Once thismid-velocity is reached, the acceleration and the jerk follows asymmetric profile until the velocity reaches V_(f) with zeroacceleration. The settling time is computed as T_(settling)=2T_(mid).

For sub-procedure 2, given a velocity model, one can compute thedistance traversed, q(t), as the area under the velocity profile, unlessan analytic expression is available.

For sub-procedure 3, it is noted that based on the theory ofderivatives, when the deviation |q(t*)−q_(linear)(t*)| reaches a maximumat some t=t*, the velocity matches the average velocity, i.e.,

${v\left( t^{*} \right)}:={{\overset{.}{q}\left( t^{*} \right)} = {{{\overset{.}{q}}_{linear}\left( t^{*} \right)} = {\frac{D}{h}.}}}$This fact is used to compute t* and |q(t*)−q_(linear)(t*)| and to decidewhether Condition 5 discussed above is satisfied or not.

Accordingly, a solution regarding the feasibility of a positiontrajectory between two points for a generic velocity model can becharacterized by the following steps.

1. For a feasibility check, one computes the settling time and settlingdistances to change the velocity from V_(entry) to V_(exit) under thegiven constraints. If the settling distance is more than the allocateddistance D, no valid duration h can be found to solve the problem.Otherwise, the problem may be solved iteratively as described herein.

2. The iteration is initiated by assuming a cruise velocity as V_(max).Then, at each iteration,

-   -   (a) compute T_(en) and T_(ex), as well as the distances        traversed during these two phases.    -   (b) If the sum of distances is more than the allocated distance,        then move to the next iteration with a reduced cruise velocity.    -   (c) If the sum of distances is less than or equal to the        allocated distance, then compute the cruise time as the ratio of        the remaining distance and the cruise velocity.    -   (d) As a final check, using Sub-Procedure 3 above, compute the        maximum deviation between the generated trajectory and a linear        trajectory over time. If the maximum deviation is within        tolerance, then one has a solution. Else, move to the next        iteration with a reduced cruise velocity.

Regarding the second problem (i.e., the optimization process, Problem2). It is noted that this problem is similar to Problem 1, except thatinstead of trying to determine the minimum duration, we want toestablish whether a given duration h is feasible while simultaneouslysatisfying the constraints expressed in Equations 6, 7, 8, 9, and 10 andto find the corresponding values of the entry settling time T_(en), theexit settling time T_(ex), the cruise duration T_(cruise) and the cruisevelocity V_(cruise)≤V_(max) for such a solution.

To address Problem 2, note that a solution herein builds on oneadditional sub-procedure where the highest and lowest possible cruisevelocities that may be attained in the given duration and satisfying allthe other kinematic constraints is calculated. This sub-procedure (i.e.,Sub-Procedure 4) calculates the highest and the lowest cruise velocitiesthat can be reached for a given allocated duration under the constraintsexpressed in Equations 6, 7, 8, 9, and 10. Note that the entry settlingtime is the time to reach some cruise velocity from a given entryvelocity and, similarly, the exit settling time is the time to reach theexit velocity from some cruise velocity under similar constraints. Themaximum effective cruise velocity V_(max_eff) is now computed byiteratively checking cruise velocities V_(max) or lower at which the sumof entry and exit settling times becomes just equal to or lower than theallocated duration. Similarly, the minimum effective cruise velocityV_(min_eff) is computed by iteratively checking cruise velocities equalto or greater than 0 at which the sum of entry and exit settling timesbecomes just equal to or lower than the allocated durationn.

FIG. 4 graphically illustrates, generally at 400, aspects of therelationships between V_(entry) (420), V_(exit) (425), V_(max), V_(min),the allocated duration, and the corresponding distances. FIG. 4 includesa graphical representation at 405 showing both the maximum velocityV_(max) (410) and the minimum velocity V_(min) (415) of zero may bereached since both the T_(en_α) (430)+T_(ex_α) (435) and T_(en_β)(440)+T_(ex_β) (445) are less than the allocated duration h (402).

In the lower graph of FIG. 4 at 412, V_(max_eff) (450)≤V_(max) (410) isthe maximum effective cruise velocity that allows T_(en_α)+T_(ex_α)=h.Similarly, V_(min_eff) (455)≥0 (V_(min), 415) is the minimum effectivecruise velocity for which T_(en_β)+T_(ex_β)=h (404). The distancestraversed are computed as the area under the velocity profiles. Here,D_(max)=D_(en_α+)D_(ex_α) and D_(min)=D_(en_β)+D_(ex_β) are the maximumand the minimum distances, respectively, that may be traversed in theallocated duration with all constraints satisfied.

Further regarding a solution to Problem 2, we herein determine,iteratively, a cruise velocity between V_(min_eff) and V_(max_eff) suchthat not only does the sum of the entry time, exit time, and cruise timeequal the allocated duration, but also the sum of the traversed distancein these phases equals the allocated distance between the two points.Referring to FIG. 4 again, the traversed distance at any phase isexpressed as the area under the velocity-time graph and the solutionfurther includes and/or considers the following:

1. Computing the settling distances (D_(en_α), D_(ex_α)) to the maximumeffective cruise velocity and the settling distances (C_(en_β),D_(ex_β)) to the minimum effective cruise velocity.

2. The maximum distance that may be traversed within the allocatedduration while reaching the maximum effective cruise velocity from thegiven entry velocity and reaching the given exit velocity from themaximum effective cruise velocity is the area under the velocity graphand may be expressed as D_(max)=D_(en_α)+D_(ex_α)+D_(cruise_α) whereD_(cruise_α)=(h−(T_(en_α)+T_(ex_α)))·V_(max_eff).

3. The minimum distance that may be required to be traversed within theallocated duration while reaching the minimum effective cruise velocityfrom the given entry velocity and reaching the given exit velocity fromthe minimum effective cruise velocity is the area under the velocitygraph and may be expressed as D_(min)=D_(en_β)+D_(ex_β)+D_(cruise_β)where D_(cruise_β)=(h−(T_(en_β)+T_(ex_β)))·V_(min_eff).

4. If D is the allocated distance and D>D_(max), then the allocatedduration is not feasible as the distance to be traveled is larger thanthe maximum that can be traveled under the constraints.

5. And, if D<D_(min), then the allocated duration is not feasible as thedistance to be traveled is smaller than the minimum that can be traveledunder the constraints.

6. If D_(min)≤D≤D_(max), we iteratively reduce the cruise velocity fromV_(max_eff) to V_(min_eff) until a cruise velocity V_(cruise) andcorresponding velocity profile v(t) are found for which the distancetraveled [area under v(t)] matches D.

7. As a final check for feasibility, a computation is made for the timeinstants t=t* where v(t*)=D/h and determine the maximum departure as|q(t*)−q_(linear)(t*)|. If it is less than Δ, the allocated duration isfeasible under the constraints.

In preparation for optimization, we compute a lower bound and an upperbound on the duration for every segment.

For the computation of the lower bound, we assume each component ismoving with maximum possible velocity and then the slowest (longest)duration is considered as the lower bound.

$\begin{matrix}{{{\forall s},j,{1 \leq s \leq M},{j = {s - 1}}}{h^{s\_\min}:={\max\limits_{\forall\xi}\frac{\left| {{\xi\left( {CP}^{j + 1} \right)} - {\xi\left( {CP}^{j} \right)}} \right.}{V_{\max}^{\xi\;,s}}}}} & (11)\end{matrix}$

For the computation of the upper bound, it is assumed that everycomponent stops at each control point. With this assumption, for a givenvelocity model, we determine the minimum duration for any component totraverse a segment while satisfying all constraints. The upper durationbound for the segment is obtained by determining the maximum of theseminimum durations. That is, ∀s, j,1≤s≤M, j=s−1 compute,

$\begin{matrix}{h^{s\_\min}:={\max\limits_{\forall\xi}{{find}\;{Min}\;{Duration}{\quad\left( {{f( \cdot )},{g( \cdot )},{\xi\left( {CP}^{j} \right)},{\xi\left( {CP}^{j + 1} \right)},0,0,V_{\max}^{\xi\;,s},A_{\max}^{\xi\;,s},J_{\max}^{\xi,\; s},\Delta^{\xi}} \right)}}}} & (12)\end{matrix}$

FIG. 5 is a flow diagram of process 500 according to some embodiments.Process 500 and the other processes described herein may be performedusing any suitable combination of hardware devices and softwareimplementations of components, devices, and systems. Software embodyingthese processes may be stored by any tangible, non-transitory medium,including but not limited to a hard disk drive, a solid-state drive, aCD-ROM, a DVD-ROM, a flash drive, optical storage, and other types ofstorage devices. The process of FIG. 5 may be implemented, in someembodiments, by at least some of the elements of system 100, yetembodiments are not limited thereto.

At operation 505, a plurality of one-dimensional positions as defined ina radiotherapy treatment plan are obtained. The plurality ofone-dimensional positions are also referred to as control points. FIG. 2introduced hereinabove may be referenced again regarding the pluralityof one-dimensional positions.

At operation 510, a set of position related constraints for a pluralityof mechanical and radiation producing components of a radiationtreatment delivery system (e.g., 100) for delivering a radiotherapytreatment plan by synchronous motion of the plurality of mechanical andradiation producing components traversing a prescribed sequence of theplurality of positions in a predetermined time is obtained. Referring toFIG. 100, the plurality of mechanical and radiation producing componentsmay include, for example, gantry 115 and radiation beam shapingcomponents therein. In accordance with aspects of the presentdisclosure, all of the all of the kinematic constraints for each of theplurality of mechanical and radiation producing components are obtainedand used in some embodiments herein. In some instances, the set ofposition related constraints for the plurality of mechanical andradiation producing components may be included in a record or set ofrecords including, at least in part, a radiotherapy treatment plan.

Process 500 proceeds to operation 515 where a determination is maderegarding a trajectory and a minimum duration for the radiationtreatment delivery system to traverse the prescribed sequence of theplurality of positions and deliver the radiotherapy treatment plan. Thetrajectory and minimum duration determined at operation 515 arecalculated while adhering to (i.e., satisfying) the constraints of eachof the plurality of mechanical and radiation producing components and,further being within a predetermined tolerance limit of a lineartrajectory along the prescribed sequence of the plurality of positions.In some embodiments, operation 515 may include iterative process(es) inorder to reach a final determination of the trajectory and minimumduration determined that satisfy the constraints of each of theplurality of mechanical and radiation producing components, includingbeing within a predetermined tolerance limit of a linear trajectoryalong the prescribed sequence of the plurality of positions.

At operation 520, the plurality of mechanical and radiation producingcomponents of the radiation treatment delivery system may be controlled(i.e., operated) to execute the determined trajectory within thedetermined minimum duration. In this manner, the radiation treatmentdelivery system may delivery radiation per the radiotherapy treatmentplan with the optimized trajectory in a minimum amount of time.

In some embodiments, an optimization process herein may be executed inan iterative manner, including for example, the following iterativeloop:

1. Allocate a duration to each segment, bounded by the lower and upperbounds. Allocation can begin by allocating shorter durations, closer tothe lower bound.

2. An average velocity of each component for each segment is calculatedby dividing the distance to be traversed by the component for thatsegment with the allocated duration of the segment.

3. The transition velocity of each component at each control point iscalculated as the mean of its average velocities in the neighboringsegments. This determines the entry and exit velocity of each componentin each segment. In some embodiments, either the harmonic mean,arithmetic mean, or some other mid values of the average velocities ofthe neighboring segments as the transition velocity may be used. Ingeneral, when an object travels the same distances with differentspeeds, its average speed is the harmonic mean of the individual speeds.Alternately, if the object travels the same duration with differentspeeds, its average speed is the arithmetic mean of the individualspeeds. However, if the component reverses direction at a control point(non-monotonic), we assign transition velocity as 0.

4. The feasibility of the duration of a segment is determined bydetermining whether the duration is feasible for every component withthe allocated entry and exit velocities, while satisfying allconstraints.

5. If feasibility is satisfied for all segments, then a solution isdetermined. Otherwise, the duration of the segment is incremented forthe failed feasibility and the process is repeated (iteratively) insearch of a feasible solution.

In some instances, when an optimized trajectory is downloaded orotherwise provided to a linac, it may be the case that the treatment isinterrupted for some reason. The point of interruption will match withsome point on the optimized trajectory in terms of position of all thecomponents. However, if the treatment is resumed from that interruptionpoint, the trajectory will not match the optimized trajectory in termsof velocity since all of the components will have to resume movementwith a zero velocity. Therefore, it is not guaranteed that all of thecomponents will reach the next control point with respective velocitiesas computed in the determining of the optimized trajectory.

However, instead of a determining a completely new trajectoryoptimization, the present disclosure includes a process to generate aresumption trajectory to quickly catch up with the pre-computed (i.e.,already determined) optimized trajectory within a few control pointsboth in terms of position and velocity.

A key aspect for the fast performance of such a process is that insteadof using the complete “remaining” trajectory plan as an input to theoptimization process, we use incremental mini-plans as input. Suchmini-plans include the point of resumption as the first control pointand a subsequent few planned control points. Also, for the last controlpoint of such a mini-plan, the components are expected to match thecorresponding control point in the optimized trajectory both in terms ofposition and velocity. Therefore, once there is a solution to theoptimization problem for the mini-plan, it can be appended to theremaining part of the pre-computed optimized trajectory to get anoverall optimized solution for resumption. Herein below, is an informaldescription of the solution.

In some embodiments, it can be assumed that the treatment interruptiontakes place between CP^(u−1) and CP^(u). Without any loss of generalitywe can assume that the radiation was on between these two CPs.Otherwise, one can run the resumption algorithm beginning with CP^(u)assuming that the linac will be setup before resumption in a way thatthe positions of all the components match that of CP^(u). Also let

_(int),

^(u−1)<

<

^(u) be the cumulative dose at the time of interruption.

The first control point of the mini-plan, namely CP_(mini) ⁰, will beconstructed with a dose value equal to the cumulative dose atinterruption. The position of the other mechanical components will bedetermined by their respective prorated value.

Given these assumptions and constraints, an iterative loop of mini-plansare constructed with incremental numbers of control points. In the firstiteration, the mini-plan consists of two control points. The firstcontrol point corresponding to the state of the components at the timeof interruption. The second is the next control point in the originalplan. In general, at every iteration, a mini-plan is constructed thatincludes one more (compared to the number of CPs in the last iteration)control points from the original plan. (1) The newly added control pointbecomes the last control point of the mini-plan of the currentiteration. Unlike the original plan, the last control point is notstatic for the mini-plan. Instead, the velocity of all the components inthe last control point of the mini-plan should match the velocity of thecomponents at the corresponding control point in the optimizedtrajectory. (2) An optimization is applied to the mini-plan.

The optimization of the mini-plan differs from the optimization of theoverall plan in the following ways. (i) Unlike the original plan, in thelast control point of the mini-plan, the components may not be static.However, for the intermediate control points of the mini-plan thetransition velocities are computed in the same way as is done in theoriginal optimization algorithm. Namely, as the arithmetic or harmonicmean of the average velocities in the neighboring segments, which inturn depend on the allocated durations. (ii) The lower bound on theduration of the segments of the mini-plan is equal to the duration ofthe corresponding segments in the optimized trajectory. However, for thefirst segment of the mini-plan, the lower bound of the duration is equalto the remaining planned duration (computed in proportion to remainingplanned dose) of the interrupted segment. (iii) The upper bound of theduration of the segments of the mini-plan is equal to the upper bound onthe duration of the corresponding segments in the optimized trajectory.However, for the first segment of the mini-plan, the upper bound iscomputed as a fraction of the original upper bound in proportion toremaining planned dose of the interrupted segment. (iv) The optimizationof the mini-plan will attempt far fewer trials than the fulloptimization. This is because, for the last control point of themini-plan, if the planned velocity of all the components cannot beachieved within a few trials, then the optimization loop enters a newiteration and adds one more control point from the original plan intothe mini plan and re-attempts optimization.

In some aspects, if a feasible allocation of durations of the segmentsof the mini-plan is found within the limited number of attempts, thenthe resumption trajectory process exits with a status of “success”. If afeasible allocation was not found and the last control point of themini-plan is not the last control point of the original plan, thenrepeat the loop. That is, the current mini-plan is augmented withanother control point from the optimized trajectory. As in the previousiteration, for this last control point, each component is notnecessarily static and is specified with the same position and velocityit held in the corresponding control point in the original plan. If afeasible allocation was not found and the last control point of themini-plan is also the last control point of the original plan, then themini-plan is now treated as a full plan and a solution is obtained withlarge number of trials, and the worst case upper bound duration of everysegment is considered.

It is noted that a large number of external beam radiotherapy treatmentplans are step-and-shoot treatment plans with alternating step-and-shootsegments. In a step segment the linac components move to prescribedpositions while the dose is switched off. In the shoot segments, thedose is delivered and the linac components maintain their prescribedpositions. Delivery of such treatment plans generally takes longer sinceeach linac component comes to a complete stop at the beginning of ashoot segment and starts again with a zero velocity at the beginning ofthe subsequent step segment.

However, dose calculation in the planning system shows that thedelivered dose to the organs may be insensitive to the departure of someof the linac components from their prescribed positions in the shootsegments, as long as such departures are within some prescribedtolerance. The tolerance that can be used for allowing some componentsto move during the shoot segment is different for different types ofcomponents. For example, for leaves, the tolerance will be quite small,but for gantry angle or other rotational components, a larger tolerancemay be possible. Such tolerances need to be decided by the planningsystem. Herein, we disclose a process that reduces the delivery time byexploiting such positional tolerance(s) while respecting constraints onthe maximum velocity, maximum acceleration and maximum jerk of allcomponents.

The underlying idea is as follows. A shoot segment is consideredmonotonic in some component provided the component is not planned tochange its direction of movement in the immediately next step segment.If a shoot segment is monotonic in some component, we may allow suchcomponent to move in the shoot segment (within tolerance of course) inits monotonic direction without worrying about the possibility that inthe immediate next step segment the component will have to change itsdirection of movement. Allowing such movement ensures that the componentneed not come to a complete stop at the beginning of a shoot segment.This aspect of a step-and-shoot treatment plan may reduce the distanceto travel for that component in the previous and the next step segment.In turn, the duration of these step segments may be reduced, providedthe component in question was the slowest component in the stepsegments. The principle is illustrated in FIG. 6. The duration for theshoot segments however does not change as they are determined by thedose to be delivered, maximum dose rate, etc. A more formal descriptionnow follows.

Referring to FIG. 6, a graphical presentation 600 shows the underlyingprinciple for converting step-and-shoot deliveries to dynamic deliveries(605), in accordance with some aspects herein. Graphic 600 includesillustrative graphs of the component velocity 610, component position615, dose rate 620, and cumulative dose 625. As shown, in a shootsegment (e.g., 655) some delivery system components may enter and exitwith non-zero velocity and the components are allowed to move 665 withina specified tolerance 670 from the original prescription. The reductionin delivery time is due to the fact that during the step segments (e.g.,650, 660), components do not necessarily start and end with zerovelocities and therefore need to travel smaller distances. It is notedthat at least some of the shoot segments (e.g., 635) may have acomponent velocity of zero for all of the components, as shown by thestatic positioning of the component(s) at 645 where the surrounding stepsegments 630 and 640 have entering and exiting velocities of zero.

Using the notations introduced hereinabove, a step-and-shoot treatmentis a treatment plan including the following properties. The treatmentcontains alternate step-and-shoot segments. In a shoot segment s betweencontrol point s and s−1, only the dose changes and the mechanicalcomponents are static, i.e., ∀ξ, ξ other than dose component

, ξ(CP^(s))=ξ(CP^(s−1)). And in a step segment, s′ between control points′ and s′−1, no dose is delivered. That is,

(CP^(s′−1))=

(CP^(s′)).

The first segment and the last segment is a shoot segment and each evennumber of segment is a step segment. Therefore, the total number ofsegments M is odd and the total number of control point M+1 is even.

Also, let s be a shoot segment between control points CP^(s) andCP^(s−1) and the prescribed position of some component ξ* in thissegment is ξ*(CP^(s))=ξ*(CP^(s−1))=: ξ*^(s). The segment s is consideredmonotonic for component ξ* if one of the following cases is satisfied:ξ*(CP^(s−2))>ξ*(CP^(s−1))=: ξ*^(s):=*(CP^(s))>ξ*(CP^(s+1)) orξ*(CP^(s−2))<ξ*(CP^(s−1))=: ξ*^(s):=ξ*(CP^(s))<ξ*(CP^(s+1))  (13)

Clearly, we may attempt to use the position tolerance of the componentin the shoot segment in order to reduce the delivery distance of in theneighboring step segments and in effect reduce the duration of the stepsegments provided (a) ξ* is the slowest component in the neighboringstep segment and (b) the shoot segment is monotonic in ξ*.

In order to avoid any bias error, while converting a shoot segment intoa dynamic segment (where the component moves while the dose is on), wewant the trajectory of the component ξ* to be equally distributed aroundthe prescribed position ξ*^(s) in the shoot segment and within theprescribed tolerance Δ^(ξ)*. Two choices are possible. Among the twochoices provided by the conversion, the one that “extends” the positionmonotonicity of the component ξ* from the neighboring step segment intoa “modified” shoot segment is used. In particular, if monotonicity isdetermined using the first case of the Equation 13 (alternatively thesecond case) in the original step-and-shoot treatment, then theconverted dynamic treatment should satisfy the first case of theEquation 14 (respectively the second case) as shown here.ξ*(T ^(s−1))=ξ*^(s)+δ^(ξ)*^(s) and ξ*(T ^(s))=ξ*^(s)−δ^(ξ)*^(s)Or ξ*(T ^(s−1))=ξ*^(s)−δ^(ξ)*^(s) and ξ*(T ^(s))=ξ*^(s)+δ^(ξ)*^(s)  (14)

In some embodiments herein, for both practical and computationalreasons, only one mechanical component ξ* (e.g., the gantry) is allowedto move, within tolerance, in the shoot segment.

Regarding the trajectory generation for step-and-shoot treatments with ashorter delivery time, we consider a step-and-shoot treatment planconsisting of M+1 control points, a velocity, acceleration and jerk foreach component ξ in each segment s,1≤s≤M and a component ξ* that isallowed to move in the shoot segment within a maximum position toleranceΔ^(ξ)* from its planned fixed value in the shoot segment, we want tofind a sequence of time points 0=T⁰≤T¹≤ . . . T^(M) such that thefollowing constraints (B1, B2, B3, B4, and B5) are simultaneouslysatisfied.

For constraint B1, all the components start with a zero velocity andacceleration at the first control point and terminate similarly at thelast control point.∀ξ{dot over (ξ)}(t=T ⁰={dot over (ξ)}(t=T ^(M))=0∀τ{umlaut over (ξ)}(t=T ⁰)={umlaut over (ξ)}(t=T ^(M))=0  (15)

For constraint B2, each of the mechanical (i.e., non-Dose) components(except ξ*) enter and exit each segment with zero velocity and remainstatic in the shoot segment. Note that segment numbers begin with 1,therefore the odd segments are shoot segments and the even segments arestep segments.∀ξ, ξ≠ξ*, ξ≠

, ∀j, 0≤j≤M, ξ(t=T ^(j))=ξ(CP^(j)), {dot over (ξ)}(t=T ^(j))=0 andodd s,1≤s≤M, ∀t,T ^(s−1) ≤t≤T ^(s), ξ(CP^(s−1))=ξ(t)=ξ(CP^(s)), {dotover (ξ)}(t)=0  (15)

Regarding constraint B3, the trajectory for the mechanical component ξ*should be such that its position will be symmetrically distributed onboth side of the prescribed position of the component in any shootsegment and will be within the position tolerance of the component. Thatis,∀odd s,1≤s≤M, ∀t,T ^(s−1) ≤t≤T ^(s), (t) is symmetrically distributedaround ξ* and |ξ*(t)−ξ*^(s)|≤Δ^(ξ)* whereξ*^(s):=ξ(CP^(s−1))=ξ(CP^(s)).  (17)

Constraint B4 states that the dose is delivered only in the shootsegments and the cumulative dose is constant at step (even) segments.That is,∀j, 0≤j≤M,

(t=T ^(j))=

(CP^(j)),

(t=T ^(j))=0 and∀ even s,1≤s≤M, T ^(s−1) ≤t≤T ^(s),

(CP^(s−1))=

(t)=

(CP^(s)),

(t)=0  (18)

Constraint B5 states that, as in the original optimization goal, foreach component ξ and for each segment s, all the constraints on velocity(V_(max) ^(ξ,s)), acceleration (A_(max) ^(ξ,s)) and jerk (J_(max)^(ξ,s)) are satisfied. That is,

$\begin{matrix}\begin{Bmatrix}{{\forall{\xi{\forall s}}},{1 \leq s \leq M},{j = {s - 1}},{\forall t},{T^{j} \leq t \leq T^{j + 1}}} \\{\left| {\overset{.}{\xi}(t)} \middle| {\leq V_{\max}^{\xi,\; s}} \right.,\left| {\overset{¨}{\xi}(t)} \middle| {\leq A_{\max}^{\xi,\; s}} \right.,\left| {\overset{...}{\xi}(t)} \middle| {\leq J_{\max}^{\xi\;,s}} \right.,}\end{Bmatrix} & (19)\end{matrix}$

It is noted that in some embodiments, an objective is to minimize thetotal delivery time T_(del) where h^(s)=T^(s)−T^(s−1)=T^(j+1)−T^(j) and

$T_{del}:={\sum\limits_{s = 1}^{M}\;{h^{s}.}}$

In some embodiments, the goals mentioned hereinabove involve thefollowing operations.

The duration of each shoot segment is solely determined by the amount ofdose to be delivered and associated constraints on maximum dose rate,rate of change of dose rate (i.e., “dose acceleration”). It may beassumed a double-S profile for dose as well. Therefore, we can determinethe duration of each shoot segment analytically as has been describedmathematically.

Next, a relationship between the entry (and the exit) velocity of thecomponent and the amount of tolerance position δ^(ξ)*^(s)≤Δ^(ξ*) ittraverses, is established. The relationship is obtained under theconstraint that the component ξ* enters and exits the shoot segment swith identical velocity and moves symmetrically in the shoot segment onboth sides of the prescribed position while staying within prescribedtolerance Δ^(ξ)*. Accordingly, the constraints on the maximum velocityand the maximum acceleration of the component in the segment is alsorespected.

Next, in every step segment the slowest and the second slowest movingcomponents (ξ_(slowest) ^(s), ξ_(second slowest) ^(s)) and thecorresponding durations (h_(slowest) ^(s), ξ_(second slowest) ^(s)) areidentified. For every step segment of the plan and for each mechanicalcomponent, we compute the minimum duration to traverse the segment underconstraints and with zero entry and exit velocity. Then, for each stepsegment, the maximum duration (and the slowest component) and secondmaximum duration (and the second slowest component) is identified asdescribed here.

$\begin{matrix}{h^{{\xi\_}s} = {{find}\;{Min}\;{Duration}{\quad{{\left( {{f( \cdot )},{g( \cdot )},{\xi\left( {CP}^{s - 1} \right)},{\xi\left( {CP}^{s} \right)},0,0,V_{\max}^{\xi,s},A_{\max}^{\xi,\; s},J_{\max}^{\xi,\; s},\Delta^{\xi\;}} \right)\mspace{79mu}\xi_{slowest}^{s}} = {{\arg\;{\max\limits_{\xi \in \Xi}\;{h^{\xi\_ s}.\mspace{79mu} h_{slowest}^{s}}}} = {{\max\limits_{\xi \in \Xi}\;{h^{\xi\_ s}.\mspace{79mu} h_{{second}\_{slowest}}^{s}}} = {\max\limits_{\xi \in {\Xi{\{\xi_{slowest}^{s}\}}}}\;{h^{\xi\_ s}.}}}}}}}} & (20)\end{matrix}$

The reason we identify the slowest and the second slowest components inthe step segments is because if the segment s+1 is a shoot segment andif in the neighboring segments s and s+2, the slowest moving componentis ξ*, i.e., ξ_(slowest) ^(s)=ξ_(slowest) ^(s+2)−ξ*, then there is thepotential opportunity to reduce the duration of the step segments s andsimilarly for the step segment s+2 by using the tolerance of thecomponent ξ* in the shoot segment s+1 as illustrated in FIG. 6. Themaximum reduction for the step segment s is limited by an amounth_(slowest) ^(s)−h_(second slowest) ^(s).

We also develop an iterative procedure to try out combinations ofreduced durations in the step segment s and increased use of positiontolerance in the shoot segment s+1 to check for a feasible profile forthe component ξ*. This procedure is repeated for every neighboringstep-and-shoot segment pair. Ultimately, a feasible combination is foundfor the component such that the duration for the step segment s isreduced and the corresponding entry/exit velocity and position of thecomponent is found in the shoot segment s+1, and the same reducedduration of the step segment s is used to generate a position profilefor all other components in that step segment.

To test some of the trajectory generation processes disclosed herein,Applicant used a RapidArc plan for a Varian system and employed thepublished constraints for the Varian TrueBeam linac with 120 leaf MLC.The plan consists of 177 control points (176 segments) and outputs385.79 MU. The mechanical constraints applied appear in Table 2. For theRapidArc plan with an exponential velocity model, the total optimizedduration was computed to be 128 s. With the double-S velocity model,where longer duration is spent in maximum acceleration, the totalduration was calculated as 98 s. The latter value is quite comparable tothe average reported duration (126 s) of RapidArc plans. It is notedthat that a simulation (e.g., a MATLAB (R2007) implementation) of theoptimized trajectory generation was executed in 70.8 s running on anIntel i7-3740QM 2.7 GHz computer with 8 GB of RAM.

TABLE 2 Mechanical constraints used for trajectory generation Max gantryspeed  6 ∘/s Max gantry acceleration  1.8 ∘/s2 Max leaf speed 25 mm/sMax leaf acceleration 520 mm/s2 Max carriage 15 mm/s Max carriageacceleration 250 mm/s2 speed Max Jaw speed 50 mm/s Max Jaw acceleration520 mm/s2 Max Dose Rate 10 MU/s Max Dose acceleration 500 MU/s2

For the same rapid arc plan, an interruption was simulated between the164th and 165th control point, the interruption occurring after deliveryof 70% of the planned segment dose of the 164th segment. For all othermechanical components (gantry, leaves, jaws), the position at the pointof interruption was computed by the same proportion of the segment dosedelivered to the total segment dose. An initial mini plan wasconstructed and the process for the resumption trajectory generation wasexecuted. The resumption process used, at most, 50 trials for each miniplan while trying to match the velocity of each component in the finalcontrol point of the mini plan with the corresponding values in theoriginal (optimized) trajectory. If such a match was not found, then themini plan was extended by another control point from the original plan.For the interrupted RapidArc plan, the resumption process was able tomatch, within 4 control points from the point of resumption, thevelocity of all components with the corresponding values in the originaltrajectory. It is noted that that a simulation (e.g., a MATLAB (R2007)implementation of the resumption process took 3.16 s to run on the samePC (i.e., Intel i7-3740QM 2.7 GHz computer with 8 GB of RAM) mentionedabove.

In some embodiments, in order to evaluate the conversion of astep-and-shoot plan into a dynamic plan, we began with a Siemens'Modulated Arc (mARC) dynamic plan. Siemens' mARC plans are rotationalIMRT (rIMRT) plans meant for burst delivery with high dose rate. Theydiffer from Varian's or Electa's VMAT plan in the sense that the dose isnot delivered while MLC leaves are moving. Instead, dose is delivered inbursts over very short arc angles and only after an MLC segment shape isformed and verified. They are dynamic plans in the sense that the gantrymoves over short arc during dose delivery. Such an rIMRT plan can beeasily converted into a step-and-shoot realization by “squeezing” thegantry spread of each shoot segment into the middle of the allowedbeam-on angular window, as is the original intention of these rIMRTplans. The resultant step-and-shoot plan consists of 71 segments. In thestep-and-shoot delivery, at the beginning of each step segment, everycomponent initiates motion with a zero velocity and comes to a completestop at the end of each step segment. In the shoot segment, only thedose is allowed to change. In a simulation by Applicant, the constraintsshown in Table 2 were used, with the exception that for gantry rotationa maximum gantry acceleration of 1°/s² was used. For the step-and-shootdelivery, the trajectory generation process herein a total delivery timeof 276.47 s was determined. As part of the conversion to dynamicsegments, the gantry is allowed to move within tolerance in the shootsegments. That is, the gantry component was considered as the componentξ* that is allowed to move in the shoot segment and a maximum tolerancevalue of Δ^(ξ)*=1.25 was allowed in the shoot segment. Consequently, thegantry never comes to a complete stop. The modified delivery schemerealized a delivery time of 236.18 s, a reduction of 15%. Anillustrative plot of the gantry position and the velocity for fewsegments is shown in FIG. 7. FIG. 7 shows the position and velocityprofiles for a gantry for a first few segments during a step-and-shootdelivery in the graphs at 705, including graphs for the gantry angle at715 and the angular speed at 720. Graphs showing, for the dynamicdelivery determined in accordance with some embodiments, are shown at710, including individual graphs for the gantry angle at 725 and theangular speed at 730.

It is reiterated that the processes and sub-procedures herein aredeveloped, at least in part, for the general class of “monotonicchange—cruise—monotonic change” velocity models. Among thesub-procedures, an important task is to find the settling time requiredto change from one velocity to another while respecting the accelerationand jerk constraints. While analytical solutions are possible fordouble-S and exponential velocity models, iterative numerical proceduresmay be used for generic velocity models. For most industrialmanipulators, velocity models such as trapezoidal, double-S and15-segment suffice. In these cases, the settling time may be obtainedanalytically. However, some of the processes disclosed herein todetermine a suitable cruise velocity to match the allocated distancewith the allocated duration are iterative in nature.

In the optimization process associated with some of the embodimentsdiscussed herein, the segment durations are the independent variables.The entry and the exit velocities of the different components at thecontrol points can also be treated as independent variables. Instead,transition velocities were chosen as either the arithmetic or theharmonic mean of the average velocities of neighboring segments. Such achoice is pragmatic and also avoids highly computationally expense.Moreover, during the optimization process, even the segment durationshave not been chosen in a completely random fashion. Instead, eachsegment durations is initialized with a value a little larger thanh^(s_min) and is increased slowly in cases where feasibility cannot beachieved.

It is noted that a main contribution of the present disclosure is toprovide an approach that works for a generic class of velocity modelsand that also respects positional tolerance constraints between viapoints. Moreover, while radiotherapy has primarily been the context orfield of application chosen for this disclosure, the processes disclosedherein are suitable for any kind of industrial application involvingsynchronous movement of multiple components.

Those in the art will appreciate that various adaptations andmodifications of the above-described embodiments can be configuredwithout departing from the scope and spirit of the claims. Therefore, itis to be understood that the claims may be practiced other than asspecifically described herein.

What is claimed is:
 1. A method for resuming a predetermined trajectoryof a radiotherapy treatment plan from an interruption point, thepredetermined trajectory including a sequence of control points of aplurality of mechanical components and radiation producing components ofa radiation treatment delivery system, each of the sequence of controlpoints including predetermined velocities of the plurality of mechanicalcomponents and radiation producing components, the method comprising:determining a mini-plan including a first mini control point and asecond mini control point, the first mini control point including theinterruption point, the second mini control point including a controlpoint of the sequence of control points that immediately follows theinterruption point; obtaining a set of position related constraints andduration related constraints for the plurality of mechanical componentsand radiation producing components, the set of position relatedconstraints and duration related constraints relating to radiationdelivery according to the radiotherapy treatment plan, the radiationdelivery involving the plurality of mechanical components and radiationproducing components traversing the sequence of control points, asdefined by the radiotherapy treatment plan; for each iteration in aniterative process including one or more iterations, determining avelocity of each of the plurality of mechanical components and radiationproducing components at the second mini control point such that theplurality of mechanical components and radiation producing componentssatisfy the set of position related constraints and duration relatedconstraints; comparing the velocity of each of the plurality ofmechanical components and radiation producing components at the secondmini control point with the predetermined velocity of each of theplurality of mechanical components and radiation producing components ofthe control point of the sequence of control points immediatelyfollowing the first mini control point to obtain a comparison result;and determining, based on the comparison result, the first mini controlpoint and the second mini control point to be used in a next iteration;and controlling the plurality of mechanical components and radiationproducing components of the radiation treatment delivery system toexecute the determined mini-plan.
 2. The method of claim 1, wherein thecomparison result includes that a rule is satisfied, and the eachiteration in an iterative process including one or more iterationsfurther includes terminating the iterative process in response to thecomparison result that the rule is satisfied.
 3. The method of claim 2,wherein the rule includes that the velocity of each of the plurality ofmechanical components and radiation producing components at the secondmini control point matches the predetermined velocity of each of theplurality of mechanical components and radiation producing components ofthe control point of the sequence of control points that immediatelyfollows the first mini control point, or that the second mini controlpoint is the last control point of the sequence of control points. 4.The method of claim 1, wherein the determining the first mini controlpoint and the second mini control point to be used in a next iterationincludes: setting the second mini control point of the current iterationas the first mini control point to be used in the next iteration; andadding one more control point of the sequence of control points thatimmediately follows the second mini control point of the currentiteration as the second mini control point to be used in the nextiteration.
 5. The method of claim 1, wherein the position relatedconstraints include, for each of the plurality of mechanical componentsand radiation producing components, at least one of a boundaryconstraint, constraints on a derivative of the position during thepredetermined trajectory, or a predetermined trajectory deviationtolerance constraint.
 6. The method of claim 5, wherein the boundaryconstraint includes at least one of an entry velocity, an exit velocity,an entry acceleration, an exit acceleration, a time taken to execute themini plan, or a distance between predetermined trajectory entry and exitpositions.
 7. The method of claim 6, wherein the boundary constraintincludes that the entry acceleration and the exit acceleration are equalto zero.
 8. The method of claim 6, wherein the boundary constraintincludes that the distance between the predetermined trajectory entryand exit positions is a predefined value.
 9. The method of claim 5,wherein the derivative of the position during the predeterminedtrajectory includes at least one of a jerk, an acceleration, or avelocity.
 10. The method of claim 1, wherein the motion of the pluralityof mechanical components and radiation producing components is definedby a velocity model having three phases including, in order, a firstmonotonic velocity phase, a constant velocity phase, and a secondmonotonic velocity phase, a direction of a velocity change in the firstmonotonic velocity phase and a direction of a velocity change in thesecond monotonic velocity phase being opposite to each other.
 11. Themethod of claim 1, wherein the interruption point is between twoconsecutive control points of the sequence of control points.
 12. Themethod of claim 1, wherein the duration related constraints include thata lower bound of a duration of a segment between the first mini controlpoint and the second mini control point of the mini-plan is equal to aduration of a corresponding segment in the predetermined trajectory, orthat an upper bound of the duration of the segment between the firstmini control point and the second mini control point of the mini-plan isequal to an upper bound of the duration of the corresponding segment inthe predetermined trajectory.
 13. The method of claim 12, wherein in thefirst iteration of the iterative process, the duration of the segmentbetween the first mini control point and the second mini control pointis equal to the duration of the corresponding segment in thepredetermined trajectory computed in proportion to a remainder of aplanned dose according to the radiotherapy treatment plan that has yetto be delivered.
 14. A system for resuming a predetermined trajectory ofa radiotherapy treatment plan from an interruption point, thepredetermined trajectory including a sequence of control points of aplurality of mechanical components and radiation producing components ofa radiation treatment delivery system, each of the sequence of controlpoints including predetermined velocities of the plurality of mechanicalcomponents and radiation producing components, the system comprising: anon-transitory computer-readable storage medium storing executableinstructions, and at least one processor in communication with thenon-transitory computer-readable storage medium, when executing theexecutable instructions, causing the system to implement a methodcomprising: determining a mini-plan including a first mini control pointand a second mini control point, the first mini control point includingthe interruption point, the second mini control point including acontrol point of the sequence of control points that immediately followsthe interruption point; obtaining a set of position related constraintsand duration related constraints for the plurality of mechanicalcomponents and radiation producing components, the set of positionrelated constraints and duration related constraints relating toradiation delivery according to the radiotherapy treatment plan, theradiation delivery involving the plurality of mechanical components andradiation producing components traversing the sequence of controlpoints, as defined by the radiotherapy treatment plan; for eachiteration in an iterative process including one or more iterations,determining a velocity of each of the plurality of mechanical componentsand radiation producing components at the second mini control point suchthat the plurality of mechanical components and radiation producingcomponents satisfy the set of position related constraints and durationrelated constraints; comparing the velocity of each of the plurality ofmechanical components and radiation producing components at the secondmini control point with the predetermined velocity of each of theplurality of mechanical components and radiation producing components ofthe control point of the sequence of control points immediatelyfollowing the first mini control point to obtain a comparison result;and determining, based on the comparison result, the first mini controlpoint and the second mini control point to be used in a next iteration;and controlling the plurality of mechanical components and radiationproducing components of the radiation treatment delivery system toexecute the determined mini-plan.
 15. The system of claim 14, whereinthe comparison result includes that a rule is satisfied, and the eachiteration in an iterative process including one or more iterationsfurther includes terminating the iterative process in response to thecomparison result that the rule is satisfied.
 16. The system of claim15, wherein the rule includes that the velocity of each of the pluralityof mechanical components and radiation producing components at thesecond mini control point matches the predetermined velocity of each ofthe plurality of mechanical components and radiation producingcomponents of the control point of the sequence of control points thatimmediately follows the first mini control point, or that the secondmini control point is the last control point of the sequence of controlpoints.
 17. The system of claim 14, wherein the determining the firstmini control point and the second mini control point to be used in anext iteration includes: setting the second mini control point of thecurrent iteration as the first mini control point to be used in the nextiteration; and adding one more control point of the sequence of controlpoints that immediately follows the second mini control point of thecurrent iteration as the second mini control point to be used in thenext iteration.
 18. The system of claim 14, wherein the interruptionpoint is between two consecutive control points of the sequence ofcontrol points.
 19. The system of claim 14, wherein the duration relatedconstraints include that a lower bound of a duration of a segmentbetween the first mini control point and the second mini control pointof the mini-plan is equal to a duration of a corresponding segment inthe predetermined trajectory, or that an upper bound of the duration ofthe segment between the first mini control point and the second minicontrol point of the mini-plan is equal to an upper bound of theduration of the corresponding segment in the predetermined trajectory.20. A non-transitory computer readable medium comprising executableinstructions for resuming a predetermined trajectory of a radiotherapytreatment plan from an interruption point, the predetermined trajectoryincluding a sequence of control points of a plurality of mechanicalcomponents and radiation producing components of a radiation treatmentdelivery system, each of the sequence of control points includingpredetermined velocities of the plurality of mechanical components andradiation producing components, when executed by at least one processor,cause the at least one processor to effectuate a method comprising:determining a mini-plan including a first mini control point and asecond mini control point, the first mini control point including theinterruption point, the second mini control point including a controlpoint of the sequence of control points that immediately follows theinterruption point; obtaining a set of position related constraints andduration related constraints for the plurality of mechanical componentsand radiation producing components, the set of position relatedconstraints and duration related constraints relating to radiationdelivery according to the radiotherapy treatment plan, the radiationdelivery involving the plurality of mechanical components and radiationproducing components traversing the sequence of control points, asdefined by the radiotherapy treatment plan; for each iteration in aniterative process including one or more iterations, determining avelocity of each of the plurality of mechanical components and radiationproducing components at the second mini control point such that theplurality of mechanical components and radiation producing componentssatisfy the set of position related constraints and duration relatedconstraints; comparing the velocity of each of the plurality ofmechanical components and radiation producing components at the secondmini control point with the predetermined velocity of each of theplurality of mechanical components and radiation producing components ofthe control point of the sequence of control points immediatelyfollowing the first mini control point to obtain a comparison result;and determining, based on the comparison result, the first mini controlpoint and the second mini control point to be used in a next iteration;and controlling the plurality of mechanical components and radiationproducing components of the radiation treatment delivery system toexecute the determined mini-plan.